Students will learn how to analyze and solve problems involving circuits with resistors in series.

### Key Equations

\begin{align*}R_{total} = R_1 + R_2 + R_3 + \ldots\end{align*}

### Guidance

**Resistors in Series:** All resistors are connected end to end. There is only one river, so they all receive the same current. But since there is a voltage drop across each resistor, they may all have different voltages across them. The more resistors in series the more rocks in the river, so the less current that flows.

#### Example 1

A circuit is wired up with two resistors in series.

Both resistors are in the same ‘river’, so both have the same current flowing through them. Neither resistor has a direct connection to the power supply so neither has 20V across it. But the combined voltages across the individual resistors add up to 20V.

**Question:** What is the total resistance of the circuit?

**Answer:** The total resistance is \begin{align*}R_{total}=R_1+R_2=90 \;\Omega+10 \;\Omega=100 \;\Omega\end{align*}

**Question:** What is the total current coming out of the power supply?

**Answer:** Use Ohm’s Law \begin{align*}(V=IR)\end{align*} but solve for current \begin{align*}(I=V/R)\end{align*}.

\begin{align*}I_{total}=\frac{V_{total}}{R_{total}}=\frac{20\:V}{100\:\Omega}=0.20\:A\end{align*}

**Question:** How much power does the power supply dissipate?

**Answer:** \begin{align*}P=IV\end{align*}, so the total power equals the total voltage multiplied by the total current. Thus, \begin{align*}P_{total}=I_{total}V_{total}=(0.20\;A)(20V)=4.0\;W\end{align*}. So the Power Supply is outputting 4W (i.e. 4 Joules of energy per second).

**Question:** How much power does each resistor dissipate?

**Answer:** Each resistor has different voltage across it, but the same current. So, using Ohm’s law, convert the power formula into a form that does not depend on voltage.

\begin{align*}P&=IV=I(IR)=I^2R.\\ P_{90 \:\Omega} &= I^2_{90\:\Omega}R_{90\:\Omega}=(0.2\:A)^2(90\:\Omega)=3.6\:W\\ P_{10 \:\Omega} &= I^2_{10\:\Omega}R_{10\:\Omega}=(0.2\:A)^2(10\:\Omega)=0.4\:W\end{align*}

\begin{align*}^*\end{align*}Note: If you add up the power dissipated by each resistor, it equals the total power outputted, as it should–Energy is always conserved.

**Question:** How much voltage is there across each resistor?

**Answer:** In order to calculate voltage across a resistor, use Ohm’s law.

\begin{align*}V_{90\:\Omega} &= I_{90\:\Omega}R_{90\:\Omega}=(0.2\:A)(90\:\Omega)=18\:V\\ V_{10\:\Omega} &= I_{10\:\Omega}R_{10\:\Omega}=(0.2\:A)(10\:\Omega)=2\:V\end{align*}

\begin{align*}^*\end{align*}Note: If you add up the voltages across the individual resistors you will obtain the total voltage of the circuit, as you should. Further note that with the voltages we can use the original form of the Power equation \begin{align*}(P=IV)\end{align*}, and we should get the same results as above.

\begin{align*}P_{90\:\Omega} &= I_{90\:\Omega}V_{90\:\Omega}=(18\:V)(0.2\:A)=3.6\:W\\ P_{10\:\Omega} &= I_{10\:\Omega}V_{10\:\Omega}=(2.0\:V)(0.2\:A)=0.4\:W\end{align*}

### Watch this Explanation

### Simulation

### Time for Practice

- Regarding the circuit below.
- If the ammeter reads \begin{align*}2\;\mathrm{A}\end{align*}, what is the voltage?
- How many watts is the power supply supplying?
- How many watts are dissipated in each resistor?

- Five resistors are wired in series. Their values are \begin{align*} 10 \Omega \end{align*}, \begin{align*} 56 \Omega \end{align*}, \begin{align*} 82 \Omega \end{align*}, \begin{align*} 120 \Omega \end{align*} and \begin{align*} 180 \Omega \end{align*}.
- If these resistors are connected to a 6 V battery, what is the current flowing out of the battery?
- If these resistors are connected to a 120 V power suppluy, what is the current flowing out of the battery?
- In order to increase current in your circuit, which two resistors would you remove?

- Given the resistors above and a 12 V battery, how could you make a circuit that draws 0.0594 A?

#### Answers to Selected Problems

- a. \begin{align*}224 \;\mathrm{V}\end{align*} b. \begin{align*}448 \;\mathrm{W}\end{align*} c. \begin{align*}400 \;\mathrm{W}\end{align*} by \begin{align*}100 \ \Omega\end{align*} and \begin{align*}48 \;\mathrm{W}\end{align*} by \begin{align*}12 \ \Omega\end{align*}
- a. 0.013 A b. 0.27 A c. \begin{align*} 120 \Omega \end{align*} and \begin{align*} 180 \Omega \end{align*}
- need about \begin{align*} 202 \Omega \end{align*} of total resistance. So if you wire up the \begin{align*} 120 \Omega \end{align*} and the \begin{align*} 82 \Omega \end{align*} in series, you'll have it.